Connes' metric for states in group algebras

• Autores: Andruchow, Esteban; Larotonda, Gabriel Andrés
• Año de publicación: 2003
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: CONNES´ METRIC; DIRAC OPERATOR; NONCOMMUTATIVE GEOMETRY; STATE
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

We follow the main idea of  A. Connes for the construction of a metric in the state space of a C*-algebra. We focus in the reduced algebra of a discrete group Г, and prove some equivalences and relations between two central objects of this category: the word-length growth (connected with the degree of the extension of Г when the group is an extension of Z), and the topological relation between the w*-topology and the one introduced with this metric in the state space of C_r*(Г). Recent studies [Antonescu] of Christensen and Antonescu show that, using a variation of the distance introduced by Connes, these topologies are equivalent if the group is of rapid decay, a concept which is equivalent in discrete groups to the concept of polynomial growth for the word-length (there is an extensive survey by Jolissant [Jol] that settles this equivalence). In this article we prove with elementary techniques, that Connes´ metric is finite and induces a topology which is equivalent to the w* topology in the state space, when the group Г is a finite extension of Z. This is not surprising at all, since M Rieffel recently established [Rieffel2] (with a complete different approach) this equivalence for Г=Z.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina